A Reliable Randomizer, Turned on Its Head

Coin tosses are a classic metaphor in economics for randomness. For instance, in his book about market efficiency, "A Random Walk Down Wall Street," economist Burton Malkiel compares the price movements of the stock market to the random outcome of a flipped coin: "[S]ometimes one gets positive price changes for several days in a row; but sometimes when you are flipping a coin you also get a long string of 'heads' in a row." According to Malkiel, mathematicians' term for a sequence of numbers produced by a random process is a random walk. To him, this is exactly what stock-price movements look like, hence the title of his book.

Similarly, Nassim Taleb, in "Fooled by Randomness," points out that the seemingly amazing success of money managers at beating the market is often best explained by pure chance. People misperceive patterns in what are, in fact, purely random sequences, akin to the outcomes of a coin flip. And as everyone knows, coin flips produce "heads and tails with 50 percent odds each."

Lately, the idea of randomness in stock prices has come under attack; prices for individual stocks (but not the market on the whole) often show small momentum effects: stocks that go up tend to keep going up, and stocks that are going down tend to keep going down. But the metaphor of a coin flip for randomness remains unquestioned. We use coin tosses to settle disputes and decide outcomes because we believe they are unbiased, with 50-50 odds.

Yet recent research into coin flips has discovered that the laws of mechanics determine the outcome of coin tosses: The startling finding is that they aren't random. Instead, for natural flips, the chance of a coin landing in the same position as it started is about 51 percent. Heads facing up predicts heads; tails predicts tails.

Three academics -- Persi Diaconis, Susan Holmes and Richard Montgomery -- made an interesting discovery through vigorous analysis at Stanford. As they note in their published results, "Dynamical Bias in the Coin Toss," the laws of mechanics govern coin flips, meaning that "their flight is determined by their initial conditions."

The physics and math behind this discovery are complex. To understand more about flips, the academics built a coin-tossing machine and filmed it using a slow-motion camera. This confirmed that the outcome of flips is not random. The machine could produce heads every time.

When people flipped the coin, results were less predictable, but there was still a slight physical bias favoring the coin's initial position 51 percent of the time. The reason real flips are less certain isn't just that the force can vary, it's that coins flipped manually tend to rotate around several axes at once. They tumble over and over, but they also spin around and around, like pizza dough being twirled. The more a coin spins, the more unpredictable the outcome.

I spoke to Holmes, one of the Stanford researchers, about this. She told me that when most people hear about this weird finding, they think it has something to do with the density of the coin, but she was able to disprove that by constructing a coin made out of balsa wood on one face and metal on the other. It made no difference come flip time. The dynamics of the coin flip, and its outcome, are determined not by the lack of balance in the coin but instead by the physics of spinning and flipping.

I asked Holmes whether coin flips used for, say, football, should be eliminated because they are biased. The answer is no, as long as the person calling the flip doesn't know how the coin is going to start out. In football, the tosser is never the caller; the tosser is supposed to be a referee. But if you are both the caller and the tosser, well, that changes things. Knowing about the bias in coin tosses give you an edge, albeit a tiny one.

Certain people, however, can make a toss come out heads (or tails) 100 percent of the time. Diaconis, Holmes's co-author and husband, is one of the people with this rare talent. Before becoming a mathematician, he was a professional magician. So how exactly is Diaconis able to make a coin toss come out a certain way? Holmes won't tell me: "It comes from his previous career -- it's magic."